Abstract

Goodness of fit tests based on sup-norm statistics of empirical processes have nonstandard limit- ing distributions when the null hypothesis is composite — that is, when parameters of the null model are estimated. Several solutions to this problem have been suggested, including the calculation of adjusted critical values for these nonstandard distributions and the transformation of the empirical process such that statistics based on the transformed process are asymptotically distribution-free. The approximation methods proposed by Durbin (1985) can be applied to compute appropriate critical values for tests based on sup-norm statistics. The resulting tests have quite accurate size, a fact which has gone unrecognized in the econometrics literature. Some justification for this accuracy lies in the similar features that Durbin’s approximation methods share with the theory of extrema for Gaussian random fields and for Gauss-Markov processes. These adjustment techniques are also related to the transformation methodology proposed by Khmaladze (1981) through the score func- tion of the parametric model. Monte Carlo experiments suggest that these two testing strategies are roughly comparable to one another and more powerful than a simple bootstrap procedure.

Item Type:

MPRA Paper

Original Title:

A comparison of alternative approaches to sup-norm goodness of git gests with estimated parameters

J. Durbin. Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. Journal of Applied Probability, 8 (3):431–453, 1971.

J. Durbin. Weak convergence of the sample distribution function when parameters are estimated. The Annals of Statistics, 1(2):279–290, 1973a.

J. Durbin. Distribution Theory for Tests Based on the Sample Distribution Function. Number 9 in Regional Conference Series in Applied Mathematics. SIAM, 1973b.

J. Durbin. Kolmogorov-Smirnov tests when parameters are estimated with applications to tests of exponentiality and tests on spacings. Biometrika, 62(1):5–22, 1975.

J. Durbin. The first-passage density of a continuous Gaussian process to a general boundary. Journal of Applied Probability, 22(1):99–122, 1985.